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Weak Goodstein sequence starting at 11.
13

%I #4 Feb 09 2015 18:26:03

%S 11,30,67,127,217,343,511,636,775,928,1095,1276,1471,1680,1903,2139,

%T 2389,2653,2931,3223,3529,3849,4183,4531,4893,5269,5659,6063,6481,

%U 6913,7359,7818,8291,8778,9279,9794,10323,10866,11423,11994,12579,13178

%N Weak Goodstein sequence starting at 11.

%C The sequence eventually goes to zero, as can be seen by noting that multiples of the highest exponent (3 in this case) only go down; in fact the 8th term, a(8) = 7*8^2 + 7*8 + 7 = 511; after which the multiple of the square term will only go down, etc.

%C This sequence, for 11, grows beyond the quintillions of digits before going to zero.

%D K. Hrbacek & T. Jech, Introduction to Set Theory, Taylor & Francis Group, 1999, pp. 125-127.

%H Harvey P. Dale, <a href="/A137411/b137411.txt">Table of n, a(n) for n = 2..1000</a>

%F To obtain a(n + 1), write a(n) in base n, increase the base to n + 1 and subtract 1.

%e a(2) = 11 = 2^3 + 2^1 + 2^0

%e a(3) = 3^3 + 3^1 + 3^0 - 1 = 30

%e a(4) = 4^3 + 4^1 - 1 = 4^3 + 3*4^0 = 67

%t nxt[{n_,a_}]:={n+1,FromDigits[IntegerDigits[a,n+1],n+2]-1}; Transpose[ NestList[ nxt,{1,11},50]][[2]] (* _Harvey P. Dale_, Feb 09 2015 *)

%Y Cf. A056004 (strong Goodstein sequences), A059933 (strong Goodstein sequence for 16.).

%K nonn

%O 2,1

%A Nicholas Matteo (kundor(AT)kundor.org), Apr 15 2008